i5-7600K vs i7-7700K.

Hyper-threading on Wikipedia.
"Hyper-Threading (HT)" is a (trademarked) Intel marketing term for their version(s) of Simultaneous Multi-Threading (SMT).
HT performance gains depend on application demand for multi-threading. Some apps (like HT benchmarks) are specifically optimized for maximal parallel-processing HT usage and (nearly) double the real performance of physical cores. Some apps require linear processing and gain nothing at all from logical multi-threading. In practice, most multi-threading apps make some middly use of HT and basically multiply overall performance of the physical cores by roughly 125% - a 4C/8T CPU is functionally a 5C/5T CPU for general real-world usage.

Games (even "heavy" games) typically require only 2 or 3 or perhaps 4 cores/threads and won't benefit from more than 4 cores (total), regardless whether the cores are logical or physical. Quad-core CPUs are the norm for all "mainstream/performance" computers - they have been the norm for many years, they'll continue to be the norm for many more - so HT (or lack of HT) will basically have no impact on gaming performance.

It turns out that HT actually requires a slightly higher transistor count, more complex circuit logic, and more power to run. All other things being equal, a CPU with HT will run just a tiny bit slower and hotter than a CPU without HT.

And it turns out that most games make awful use of HT. A few games are notorious for performing worse with HT than without.

The i7-7700K is a higher-binned and faster-clocked CPU. With higher overclocking potential. De-lidding means hard overclocking, and it seems wasteful (to me) to do it with the lesser i5 CPU.

Gaming 144Hz @ 1440p means up to 144fps @ 1440p. This is all about the GPU(s), only barely about the CPU. And both of these CPUs has (more than) enough cores/threads to run games. I'd say your "cooked" 5.1GHz i7-7700K is already better for gaming than any i5-7600K, especially since HT is of no real consequence, although who knows what other functions may have also been cooked?

I'd say that if you're gonna de-lid and overclock then i7-7700K. Otherwise, keep running your "cooked" i7-7700K until it craps out, it might continue to play your games for a long time while processor prices keep dropping (and while, maybe, newer/faster processors are released).

If you're balancing CPU options based on price, keep running your existing CPU (or get the cheaper i5, if you must) and spend their rest of your budget towards upgrading your GPU card to the most pwnage you can afford.